Finite Mathematics Quiz: Matrix Operations

13 questions · exam conditions

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Matrix OperationsQuestion 1 of 13

A transformation matrix $T = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ satisfies $T \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 3 \\ -2 \end{pmatrix}$ and $T \begin{pmatrix} 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 1 \\ 4 \end{pmatrix}$ . What is the value of $\det(2T)$ ?

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Finite Mathematics Quiz

Finite Mathematics Quiz: Matrix Operations

Practice Matrix Operations in Finite Mathematics with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Matrix Operations, giving you a quick way to practice the rules, question types, and explanations that matter most for Finite Mathematics.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

56 (correct answer)
28
14
42

Explanation: From the given conditions, T(1,0) = (3,-2) gives us the first column of T is (3,-2). T(0,1) = (1,4) gives us the second column of T is (1,4). Therefore T = (3 1; -2 4). The determinant of T is (3)(4) - (1)(-2) = 12 + 2 = 14. For matrix 2T, det(2T) = 2² × det(T) = 4 × 14 = 56.

Question 2

Let matrices $A = \begin{pmatrix} 1 & 5 \\ -2 & 3 \end{pmatrix}$ and $B = \begin{pmatrix} 2 & -1 \\ 1 & 0 \end{pmatrix}$ . If $X$ is a $2 \times 2$ matrix such that $2X + A = B^2$ , which of the following is matrix $X$ ?

$\begin{pmatrix} 1 & -3.5 \\ 2 & -2 \end{pmatrix}$ (correct answer)
$\begin{pmatrix} -1 & 3.5 \\ -2 & 2 \end{pmatrix}$
$\begin{pmatrix} 1.5 & -2 \\ 1.5 & -1.5 \end{pmatrix}$
$\begin{pmatrix} 2 & 1.5 \\ 0 & 1 \end{pmatrix}$

Explanation: The problem requires solving the matrix equation

2X + A = B^2

for

X

. First, rearrange the equation to isolate

X

X = \frac{1}{2}(B^2 - A)

. Step 1: Calculate

B^2

Step 2: Calculate

B^2 - A

Step 3: Multiply by the scalar

\frac{1}{2}

Distractor B comes from incorrectly solving for

X

X = \frac{1}{2}(A - B^2)

, a sign error. Distractor C results from incorrectly calculating

B^2

by squaring each element of

B

. Distractor D arises from incorrectly calculating

X = \frac{1}{2}(B^2 + A)

Question 3

Let $A$ be a $3 \times 2$ matrix, $B$ be a $2 \times 3$ matrix, $C$ be a $3 \times 3$ matrix, and $D$ be a $2 \times 2$ matrix. Which of the following matrix expressions is defined?

$A(B+C)$
$C^2 - B$
$D + BA$ (correct answer)
$AC + D$

Explanation: To determine if a matrix expression is defined, we must check the dimensions for each operation. A) For

A(B+C)

, the addition

B+C

requires matrices of the same dimensions.

B

2 \times 3

and

C

3 \times 3

, so they cannot be added. The expression is undefined. B) For

C^2 - B

, the operation

C^2 = C \times C

is defined because

C

is square (

3 \times 3

). The result is a

3 \times 3

matrix. However, to subtract

B

, which is

2 \times 3

, the dimensions must match. Since

3 \times 3 \neq 2 \times 3

, the subtraction is not possible. The expression is undefined. C) For

D + BA

, first consider the product

BA

. The inner dimensions must match.

B

2 \times 3

and

A

3 \times 2

. The inner dimensions (3 and 3) match, so

BA

is defined. The resulting matrix

BA

has dimensions

2 \times 2

. The addition

D + BA

is defined because both

D

and

BA

are

2 \times 2

matrices. This expression is defined. D) For

AC+D

, consider the product

AC

A

3 \times 2

and

C

3 \times 3

. The inner dimensions (2 and 3) do not match, so the product

AC

is undefined. The entire expression is undefined.

Question 4

A company sells three products at two different stores. The quantity of each product sold at each store is given by the matrix $Q$ . The wholesale cost and retail price for each product are given by the matrix $P$ .

In matrix $Q$ , the rows represent Store 1 and Store 2, respectively, and the columns represent Products 1, 2, and 3. In matrix $P$ , the rows correspond to the products, and the columns represent wholesale cost and retail price, respectively.

Based on the information, what is the total profit for Store 2?

$3,700
$3,740 (correct answer)
$9,900
$13,640

Explanation: To find the total profit, we first need to calculate the total wholesale cost and total retail revenue for each store. This can be done by computing the matrix product

QP

. The resulting matrix will have rows representing stores and columns representing total cost and total revenue.

(QP)_{11} = 100(10) + 150(20) + 200(30) = 1000 + 3000 + 6000 = 10000

(QP)_{12} = 100(15) + 150(28) + 200(40) = 1500 + 4200 + 8000 = 13700

(QP)_{21} = 120(10) + 180(20) + 170(30) = 1200 + 3600 + 5100 = 9900

(QP)_{22} = 120(15) + 180(28) + 170(40) = 1800 + 5040 + 6800 = 13640

So, the resulting matrix is

QP = \begin{pmatrix} 10000 & 13700 \\ 9900 & 13640 \end{pmatrix}

. The second row corresponds to Store 2. The total wholesale cost for Store 2 is $9,900, and the total retail revenue is $13,640. The profit is the difference: Profit = Revenue - Cost = $13,640 - $9,900 = $3,740. Distractor A is the profit for Store 1 ($13,700 - $10,000 = $3,700). Distractor C is the total wholesale cost for Store 2. Distractor D is the total retail revenue for Store 2.

Question 5

Given matrices $A = \begin{pmatrix} 1 & 0 \\ 3 & 2 \end{pmatrix}$ and $B = \begin{pmatrix} -1 & 4 \\ 1 & 0 \end{pmatrix}$ , what is the result of the operation $(A + B^T)^2$ ?

$\begin{pmatrix} 7 & 2 \\ 14 & 11 \end{pmatrix}$ (correct answer)
$\begin{pmatrix} 16 & 8 \\ 8 & 20 \end{pmatrix}$
$\begin{pmatrix} 6 & -1 \\ 5 & 8 \end{pmatrix}$
$\begin{pmatrix} 0 & 1 \\ 49 & 4 \end{pmatrix}$

Explanation: This is a multi-step problem involving matrix transpose, addition, and multiplication. Step 1: Find the transpose of

B

, denoted

B^T

Step 2: Add

A

and

B^T

Step 3: Square the resulting matrix.

Distractor B is the result of computing

(A+B)^2

, failing to take the transpose of

B

. Distractor C is the result of incorrectly assuming that

(X+Y)^2 = X^2 + Y^2

for matrices and calculating

A^2+(B^T)^2

. Distractor D is the result of squaring each element of

(A+B^T)

instead of performing matrix multiplication.

Question 6

Consider the matrices $A = \begin{pmatrix} 1 & 0 & 2 \\ -1 & 3 & 1 \end{pmatrix}$ , $B = \begin{pmatrix} 1 & 0 & 1 & 2 \\ 4 & 1 & 0 & -1 \\ 0 & 2 & 3 & 1 \end{pmatrix}$ , and $C = \begin{pmatrix} 5 & 1 \\ 0 & 2 \\ -1 & 0 \\ 2 & 3 \end{pmatrix}$ . What is the entry in the second row, first column of the product matrix $ABC$ ?

9
21
45 (correct answer)
61

Explanation: We need to find the entry

(ABC)_{21}

. This can be found without computing the entire matrix product. Let

D = AB

. The entry

(ABC)_{21}

is the same as

(DC)_{21}

, which is the dot product of the second row of

D

and the first column of

C

. Step 1: Find the second row of

D = AB

. The second row of

AB

is found by multiplying the second row of

A

by the matrix

B

This results in a

1 \times 4

row vector:

\begin{pmatrix} -1(1)+3(4)+1(0) & -1(0)+3(1)+1(2) & -1(1)+3(0)+1(3) & -1(2)+3(-1)+1(1) \end{pmatrix} $$ $$ = \begin{pmatrix} 11 & 5 & 2 & -4 \end{pmatrix}

This is the second row of the matrix

D = AB

. Step 2: Find the dot product of this row with the first column of

C

$$ $$ = 11(5) + 5(0) + 2(-1) + (-4)(2) = 55 + 0 - 2 - 8 = 45$$ Thus, the entry $(ABC)_{21}$ is 45. Distractor A, 9, is $(ABC)_{22}$. Distractor B, 21, is $(ABC)_{12}$. Distractor D, 61, is a calculation error, likely a sign error in the final step ($55-2+8$).

Question 7

Let $A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ and $I$ be the $2 \times 2$ identity matrix. If $B = I + A$ , what is the matrix $B^5$ ?

$\begin{pmatrix} 1 & 5 \\ 0 & 1 \end{pmatrix}$ (correct answer)
$\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$
$\begin{pmatrix} 5 & 5 \\ 0 & 5 \end{pmatrix}$
$\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$

Explanation: This problem can be solved by observing a pattern in the powers of

B

. First, note the properties of matrix

A

A^2 = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} = Z

(the zero matrix). All higher powers

A^k

for

k \ge 2

will also be the zero matrix. Now, let's compute the first few powers of

B = I+A

. Since

I

and

A

commute (

IA=AI=A

), we can use the binomial expansion.

B^2 = (I+A)^2 = I^2 + 2IA + A^2 = I + 2A + Z = I + 2A

B^3 = (I+A)^3 = I^3 + 3I^2A + 3IA^2 + A^3 = I + 3A + 3Z + Z = I + 3A

. The pattern is

B^n = I + nA

. Therefore, for

n=5

Distractor B is

B

itself. Distractor C represents $5B$. Distractor D is the identity matrix, which might be chosen if one mistakenly thought

(I+A)

somehow simplifies to

I

Question 8

Let $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$ . If the matrices commute, meaning $AB = BA$ , which of the following conditions on the entries of $A$ must be true?

$a=b$ and $c=d$
$a=c$ and $b=d$
$a=d$ and $b=c$ (correct answer)
$a+d = b+c$

Explanation: To find the condition for commutativity, we compute both products

AB

and

BA

and set them equal to each other. First, compute

AB

Next, compute

BA

Now, set

AB = BA

\begin{pmatrix} a+b & a+b \\ c+d & c+d \end{pmatrix} = \begin{pmatrix} a+c & b+d \\ a+c & b+d \end{pmatrix}

For these matrices to be equal, their corresponding entries must be equal. This gives a system of equations:

$a+b = a+c \implies b=c$
$a+b = b+d \implies a=d$
$c+d = a+c \implies d=a$
$c+d = b+d \implies c=b$ All four equations lead to the same two conditions: $a=d$ and $b=c$ . This means the diagonal elements must be equal, and the off-diagonal elements must be equal. Distractors A and B represent other forms of symmetry in the matrix that are not the required condition. Distractor D is a condition related to the trace, but it is not sufficient for commutativity with this specific matrix $B$ .

Question 9

A manufacturing process involves two stages. Stage 1 converts two types of raw materials into three types of intermediate goods. Stage 2 converts the intermediate goods into two types of final products. The amounts are described by matrices $R$ and $P$ . Matrix $R = \begin{pmatrix} 10 & 5 & 8 \\ 4 & 6 & 12 \end{pmatrix}$ shows the units of raw materials (rows 1 and 2) needed for each intermediate good (columns 1, 2, and 3). Matrix $P = \begin{pmatrix} 50 & 60 \\ 20 & 10 \\ 5 & 30 \end{pmatrix}$ shows the units of intermediate goods (rows 1, 2, and 3) needed for each final product (columns 1 and 2).

How many units of Raw Material 2 are required to produce one unit of Final Product 1?

240
380 (correct answer)
640
660

Explanation: To find the number of units of each raw material needed for each final product, we need to multiply the matrices

R

and

P

. The resulting matrix,

M = RP

, will show the units of raw material

i

(row) needed for final product

k

(column).

We need the entry corresponding to Raw Material 2 (row 2) and Final Product 1 (column 1), which is

M_{21}

M_{21} = (\text{row 2 of } R) \cdot (\text{column 1 of } P)

M_{21} = (4)(50) + (6)(20) + (12)(5) = 200 + 120 + 60 = 380

So, 380 units of Raw Material 2 are needed for one unit of Final Product 1. Distractor A, 240, comes from calculating the product in the wrong order,

PR

, which is not meaningful in this context. Distractor C, 640, is the amount of Raw Material 1 needed for Final Product 1 (

M_{11}

). Distractor D, 660, is the amount of Raw Material 2 needed for Final Product 2 (

M_{22}

Question 10

Let $A = \begin{pmatrix} 1 & 0 \\ 2 & 1 \end{pmatrix}$ , $B = \begin{pmatrix} 0 & 1 \\ -1 & 2 \end{pmatrix}$ , and $C = \begin{pmatrix} 2 & 3 \\ 1 & 8 \end{pmatrix}$ . If $x$ and $y$ are scalars such that $xA + yB = C$ , what is the value of $x+y$ ?

-1
1
5 (correct answer)
There are no such scalars $x$ and $y$ .

Explanation: The matrix equation

xA + yB = C

translates into a system of linear equations for the scalars

x

and

y

Performing the scalar multiplication and addition on the left side gives:

\begin{pmatrix} x & 0 \\ 2x & x \end{pmatrix} + \begin{pmatrix} 0 & y \\ -y & 2y \end{pmatrix} = \begin{pmatrix} x & y \\ 2x-y & x+2y \end{pmatrix}

Setting this equal to matrix

C

gives a system of four equations:

$x = 2$
$y = 3$
$2x - y = 1$
$x + 2y = 8$ From the first two equations, we get the potential solution $x=2$ and $y=3$ . We must check if these values satisfy the other two equations. For equation 3: $2(2) - 3 = 4 - 3 = 1$ . This is true. For equation 4: $2 + 2(3) = 2 + 6 = 8$ . This is also true. Since the solution $x=2, y=3$ is consistent for all entries, the scalars exist. The question asks for the value of $x+y$ , which is $2+3=5$ . Distractors A and B result from errors in solving the system. Distractor D would be correct if the system were inconsistent.

Question 11

Let $A = \begin{pmatrix} 1 & 2 \\ 3 & -1 \end{pmatrix}$ and $B = \begin{pmatrix} 2 & 0 \\ 1 & 4 \end{pmatrix}$ . What is the matrix product $AB - BA$ ?

$\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$
$\begin{pmatrix} 6 & 12 \\ 18 & -6 \end{pmatrix}$
$\begin{pmatrix} -2 & -4 \\ 8 & 2 \end{pmatrix}$
$\begin{pmatrix} 2 & 4 \\ -8 & -2 \end{pmatrix}$ (correct answer)

Explanation: This question tests matrix multiplication and subtraction, and highlights that matrix multiplication is not commutative (

AB \neq BA

). First, compute

AB

Next, compute

BA

Finally, compute the difference

AB - BA

Distractor A is chosen by students who incorrectly assume matrix multiplication is commutative, making

AB - BA

the zero matrix. Distractor B is the result of adding

AB+BA

. Distractor C is the result of

BA - AB

, which is a sign error.

Question 12

Let $A$ and $B$ be $n \times n$ matrices. If the equation $(A+B)^2 = A^2 + 2AB + B^2$ holds true, which of the following statements must be true?

A or B must be the zero matrix.
A or B must be the identity matrix.
$A$ and $B$ must be invertible.
$A$ and $B$ must commute. (correct answer)

Explanation: This question tests the understanding of matrix multiplication properties, specifically its non-commutativity. Let's expand the left side of the equation:

(A+B)^2 = (A+B)(A+B) = A(A+B) + B(A+B) = A^2 + AB + BA + B^2

Now, set this expanded form equal to the right side of the given equation:

A^2 + AB + BA + B^2 = A^2 + 2AB + B^2

Subtract

A^2

and

B^2

from both sides:

AB + BA = 2AB

Subtract

AB

from both sides:

BA = AB

This final equation is the definition of commutativity for matrices

A

and

B

. Therefore,

A

and

B

must commute. Distractors A and B are sufficient conditions for the equation to hold, but they are not necessary. For example, if

A = 2I

and

B = 3I

, neither is the zero matrix, but they commute, and the equation holds. Distractor C is irrelevant; invertibility is not required for the property to hold.

Question 13

Let $A = \begin{pmatrix} 3 & 1 \\ 4 & 2 \end{pmatrix}$ and $I$ be the $2 \times 2$ identity matrix. Find the value of the scalar $k$ such that $A^2 - 5A = kI$ .

-5
-2 (correct answer)
2
5

Explanation: This problem involves matrix multiplication, scalar multiplication, and subtraction to solve for an unknown scalar. Step 1: Compute

A^2

Step 2: Compute

5A

Step 3: Compute the left side of the equation,

A^2 - 5A

Step 4: Set the result equal to

kI

and solve for

k

\begin{pmatrix} -2 & 0 \\ 0 & -2 \end{pmatrix} = \begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}

By comparing the matrices, we see that

k = -2

. Distractor C comes from a sign error or from using the determinant of

A

, which is

3(2)-1(4)=2

. Distractors A and D come from misinterpreting the scalar in the equation.